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A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations. (English) Zbl 0756.65129
The authors propose a new splitting technique in which the computation of the fractional steps is independent of the previous fractional steps. The method can therefore be implemented on parallel computers. Convergence estimates for the scheme proposed are given for steady state and nonsteady state linear and nonlinear problems.
In detail, the paper deals with linear and nonlinear elliptic problems, with applications of the parallel splitting method to steady Navier- Stokes-problems, with splitting methods for linear and quasilinear evolution equations, and with parallel splitting methods for the evolution Navier-Stokes equations. The paper is mainly concerned with the numerical aspects of the problems indicated.
Reviewer: E.Krause (Aachen)

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65Y05 Parallel numerical computation
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35Q30 Navier-Stokes equations
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
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References:
[1] A. J. CHORIN, Numerical solution of Navier-Stokes equations Math. Comp., 22 (1968), 745-762. Zbl0198.50103 MR242392 · Zbl 0198.50103 · doi:10.2307/2004575
[2] C. CUVELIER, A. SEGAL, A. A. VAN STEENHOVEN, Finite element methods and Navier-Stokes equations, D. Reidel Publishing Company, 1986. Zbl0649.65059 MR850259 · Zbl 0649.65059
[3] J. DOUGLAS and H. RACHFORD, On the numencal solution of the heat conduction problem in two and three space variables, Trans. Amer. Math. Soc., 82 2 (1956), 421-439. Zbl0070.35401 MR84194 · Zbl 0070.35401 · doi:10.2307/1993056
[4] G. GiRAULT and P.-A. RAVIART, Finite element methods for Navier-Stokes equations, Theory and algorithms, Springer-Verlag, Berlin, 1986. Zbl0585.65077 MR540128 · Zbl 0585.65077
[5] A. R. GOURLAY, Splitting-up methods for time dependent partial differential equations, in The state of the art in numerical analysis (proc. Conf. Univ. York, Heslington, 1976), Academic Press, London, 1977, pp. 757-796. MR451759
[6] J. G. HEYWOOD and R. RANNACHER, Finite element approximation of the nonstationary Navier Stokes problem : I. Regularity of the solution and second-order error estimates for the spatial discretization, SIAM J. Numer. Anal., 19 (1982), 275-311. Zbl0487.76035 MR650052 · Zbl 0487.76035 · doi:10.1137/0719018
[7] M. KŘIŽEK and P. NEITTAANMÄKI, Finite element approximation to variational problems with applications, Pitman Monographs in Pure and Applied Mathematics 50, Longman, 1990. Zbl0708.65106 MR1066462 · Zbl 0708.65106
[8] O. A. LADYZHENSKAYA, The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, Second Edition, 1969. Zbl0184.52603 MR254401 · Zbl 0184.52603
[9] O. A. LADYZHENSKAYA and V. I. RIVKIND, On the alternating method for the computation of a viscous incompressible fluid flow in cylindrical coordinates, Izv. Akad. Nauk, 35 (1971), 251-268. Zbl0222.76024 · Zbl 0222.76024
[10] J. L. LIONS and R. TEMAM, Éclatement et décentralisation en calcul des variations, in < Proc. of Symposium on Optimization > , Lecture Notes in Mathematics, Vol. 132, Springer Verlag, 1970, pp. 196-217. Zbl0223.49033 MR467468 · Zbl 0223.49033
[11] T. LU, P. NEITTAANMÄKI and X.-C. TAI, A parallel spliting-up method and its application to Naver-Stokes equations, Appl. Math. Lett., 4 (1989). 25-29. Zbl0718.65066 MR1095644 · Zbl 0718.65066 · doi:10.1016/0893-9659(91)90161-N
[12] G. I. MARCHUK, Methods of numerical mathematics, Springer-Verlag, 1982. Zbl0485.65003 MR661258 · Zbl 0485.65003
[13] D. W. PEACEMAN and H. H. RACHFORD, The numerical solution of parabolic and elliptic differential equations, SIAM, 3 (1955), 28-42. Zbl0067.35801 MR71874 · Zbl 0067.35801 · doi:10.1137/0103003
[14] J. SHEN, On error estimates of projection methods for Navier-Stokes equations : First order schemes, To appear in SIAM J. Numer. Anal. Zbl0741.76051 MR1149084 · Zbl 0741.76051 · doi:10.1137/0729004
[15] X. C. TAI and P. NEITTAANMÄKI, A parallel finite element splitting up method for parabolic problems, Numerical methods for partial differential equations, 7 (1991), 209-225. Zbl0747.65084 MR1122113 · Zbl 0747.65084 · doi:10.1002/num.1690070302
[16] R. TEMAMSur l’approximation de la solution des équations de Navier-Stokes par la méthode de pas fractionnaires (I), Arch. Rational Mech. Anal., 32 (1969), 135-153. Zbl0195.46001 MR237973 · Zbl 0195.46001 · doi:10.1007/BF00247678
[17] R. TEMAM, Numerical analysis, D. Reidel Publishing Company, Dordrecht, North-Holland, 1973. Zbl0261.65001 MR347099 · Zbl 0261.65001
[18] R. TEMAMNavier-Stokes equations, North-Holland, 1977. Zbl0383.35057 MR609732 · Zbl 0383.35057
[19] N. N. YANENKO, The method of fractional steps for solving multi-dimensional problems of mathematical physics, Novosibirsk, Nauka, 1967. Zbl0209.47103 · Zbl 0209.47103
[20] L. YING, Viscosity splitting method for three dimensional Navier-Stokes equations, Acta Math. Sinica New Series, No. 3, 4 (1988), 210-226. Zbl0673.35085 MR965569 · Zbl 0673.35085 · doi:10.1007/BF02560577
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